Evaluate: sqrt(x+3)=sqrt(2x+8)-1

Expression: $\sqrt{ x+3 }=\sqrt{ 2x+8 }-1$

Square both sides of the equation

$x+3=2x+8-2\sqrt{ 2x+8 }+1$

Add the numbers

$x+3=2x+9-2\sqrt{ 2x+8 }$

Move the expression to the left-hand side and change its sign

$x+3+2\sqrt{ 2x+8 }=2x+9$

Move the expression to the right-hand side and change its sign

$2\sqrt{ 2x+8 }=2x+9-x-3$

Collect like terms

$2\sqrt{ 2x+8 }=x+9-3$

Subtract the numbers

$2\sqrt{ 2x+8 }=x+6$

Square both sides of the equation

$4\left( 2x+8 \right)={x}^{2}+12x+36$

Distribute $4$ through the parentheses

$8x+32={x}^{2}+12x+36$

Move the expression to the left-hand side and change its sign

$8x+32-{x}^{2}-12x-36=0$

Collect like terms

$-4x+32-{x}^{2}-36=0$

Calculate the difference

$-4x-4-{x}^{2}=0$

Factor out the negative sign from the expression and reorder the terms

$-\left( {x}^{2}+4x+4 \right)=0$

Use ${a}^{2}+2ab+{b}^{2}={\left( a+b \right)}^{2}$ to factor the expression

$-{\left( x+2 \right)}^{2}=0$

Change the signs on both sides of the equation

${\left( x+2 \right)}^{2}=0$

The only way a power can be $0$ is when the base equals $0$

$x+2=0$

Move the constant to the right-hand side and change its sign

$x=-2$

Check if the given value is the solution of the equation

$\sqrt{ -2+3 }=\sqrt{ 2 \times \left( -2 \right)+8 }-1$

Simplify the expression

$1=1$

The equality is true, therefore $x=-2$ is a solution of the equation

$x=-2$